Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2017 / Article

Research Article | Open Access

Volume 2017 |Article ID 6043109 | 15 pages | https://doi.org/10.1155/2017/6043109

Estimation of Distribution Algorithm Using Correlation between Binary Elements: A New Binary-Code Metaheuristic

Academic Editor: Benjamin Ivorra
Received29 Apr 2017
Revised28 Jul 2017
Accepted02 Aug 2017
Published13 Sep 2017

Abstract

A new metaheuristic called estimation of distribution algorithm using correlation between binary elements (EDACE) is proposed. The method searches for optima using a binary string to represent a design solution. A matrix for correlation between binary elements of a design solution is used to represent a binary population. Optimisation search is achieved by iteratively updating such a matrix. The performance assessment is conducted by comparing the new algorithm with existing binary-code metaheuristics including a genetic algorithm, a univariate marginal distribution algorithm, population-based incremental learning, binary particle swarm optimisation, and binary simulated annealing by using the test problems of CEC2015 competition and one real-world application which is an optimal flight control problem. The comparative results show that the new algorithm is competitive with other established binary-code metaheuristics.

1. Introduction

Nowadays in the economic-competitive world, optimisation has become increasingly popular for real applications as it is a powerful mathematical tool for solving a wide range of engineering design types. Once an optimisation problem is posed, one of the most important elements in the optimisation process is an optimisation method or an optimiser used to find the optimum solution. Optimisers can be categorised as the methods with and without using function derivatives. The former are traditionally called mathematical programming or gradient-based optimisers whereas the latter have various subcategories. One of them is a metaheuristic (MH). The term metaheuristics can cover nature-inspired optimisers [110], swarm intelligent algorithms [1120], and evolutionary algorithms [2124]. Most of them are based on using a set of design solutions, often called a population, for searching an optimum. The main operator usually consists of the reproduction and selection stages. The advantages of such an optimiser are simplicity to use, global optimisation capability, and flexibility to apply as it is derivative-free. However, it still has a slow convergence rate and search consistency. These issues have made researchers and engineers around the globe investigate how to improve the search performance of MHs.

A genetic algorithm (GA) [21] is probably the best known MH while other popular methods are differential evolution (DE) [22] and particle swarm optimisation (PSO) [17]. Among MH algorithms, they can be categorised as the methods using real, binary, or integer codes. The mix of those types of design variables and some other types can also be made. This makes MHs considerably appealing for use with real-world applications particularly for those design problems that function derivatives are not available or impossible to calculate. Most MHs are based on continuous design variables or real codes. For single objective optimisation, there have been numerous real-code MHs being developed. At the early stage, methods like evolutionary programming [25] and evolution strategies [26] were proposed. Then, DE and PSO were introduced. Until recently, there have been probably over a hundred new real-code MHs in the literature. Some recent algorithms include, for example, a sine-cosine algorithm [27], a grey wolf optimiser [20], teaching-learning-based optimisation [2], and Jaya algorithm [28]. Meanwhile, powerful existing algorithms such as PSO and DE have been upgraded by integrating into them some types of self-adaptive schemes, for example, adaptive differential evolution with optional external archive (JADE) [29], Success-History Based Parameter Adaptation for Differential Evolution (SHADE) [30], SHADE Using Linear Population Size Reduction (LSHADE) [31], and adaptive PSO [3234]. MHs are even more popular when they can be used to find a Pareto front of a multiobjective optimisation problem within one optimisation run. Such a type of algorithm is usually called multiobjective evolutionary algorithms (MOEAs) where some of the best known algorithms are nondominated sorting genetic algorithm (NSGA-I, NSGA-II, and NSGA-III) [3537], multiobjective particle swarm optimisation [38], strength Pareto evolutionary algorithm [39], multiobjective grey wolf optimisation [40], multiobjective teaching-learning-based optimisation [41], multiobjective evolutionary algorithm based on decomposition [42], multiobjective ant colony optimisation [43], multiobjective differential evolution [44], and so forth. One of the most challenging issues in MHs is to improve their ability for tackling many-objective optimisation (a problem with more than three objectives). Some recently proposed algorithms are knee point-driven evolutionary algorithm [45], an improved two-archive algorithm [46], preference-inspired coevolutionary algorithms [47], and so forth.

In practice, GA a metaheuristic using binary strings is arguably the most used method as it is included in engineering software such as MATLAB. Apart from GA, other MHs using a binary string representing a design solution include a univariate marginal distribution algorithm (UMDA) [48], population-based incremental learning (PBIL) [24], binary particle swarm optimisation (BPSO) [49], binary simulated annealing (BSA) [50], binary artificial bee colony algorithm based on genetic operator (GBABC) [51], binary quantum-inspired gravitational search algorithm (BQIGSA) [52], and self-adaptive binary variant of a differential evolution algorithm (SabDE) [53]. With the popularity of GA, a binary-code MH has been rarely developed and proposed while its real-code counterparts have over a hundred different search concepts reported in the literature. That means there are possible more than a thousand real-code MH algorithms being published. It should be noted that real-code MHs can be modified to solve binary-code optimisation by means of binarisation [54].

This paper is therefore devoted to the further development of a binary-code metaheuristic. The method is called estimation of distribution algorithm using correlation between binary elements (EDACE). Performance assessment is made by comparing the proposed optimiser with GA, UMDA, BPSO, BSA, and PBIL by using the CEC2015 test problems. Also, the real-world optimal flight control is used for the assessment. The comparative results are obtained and discussed. It is shown that EDACE is among the top performers.

2. Proposed Method

The simplest but efficient estimation of distribution algorithm is probably population-based incremental learning (PBIL). Another MH that uses a similar concept is UMDA. Unlike GA which uses a matrix containing the whole binary solutions during the search, PBIL uses the so-called probability vector to represent a binary population. During an optimisation process, the probability vector is updated iteratively until approaching an optimum. In EDACE, a matrix called a correlation between binary elements (CBE) matrix is used to represent a binary population. The matrix can be denoted as , where the value of the element indicates the correlation between element and element of a binary design solution. The higher value of means the higher probability that binary elements and will have the same value. The algorithm is developed to deal with a box-constrained optimisation problem:where is an objective function and is a vector containing design variables (a design vector). and are the lower and upper bounds of , respectively. Assuming that a design vector can be represented by a row vector of binary bits size , the CBE matrix thus has the size of . It should be noted that the details of converting a binary string to be a design vector can be found in [55]. In generating a binary string from the CBE matrix, a reference binary solution (RBS) is needed. It can be a randomly generated solution or the best solution found so far depending on a user preference. Then, a row of the matrix is randomly selected (say the th row). The th element of a generated binary solution is set to be the th element of the reference binary solution. The rest of the created binary elements are based on the value of ; . The procedure for creating a binary solution sized from the CBE matrix is detailed in Algorithm 1 where is a binary design solution, is the reference binary solution, is a population size, and is a uniform random number. The algorithm spends loops for creating binary solutions. The process for generating a binary solution from the CBE matrix is in steps . For one binary solution, only one randomly selected row of CBE (say row ) is used (step ). Then, the th element of a generated binary solution is set equal to the th element of the reference binary solution, . The rest of the elements of the generated binary solution are created in such a way that their values depend on corresponding elements on the th row of CBE. From the computation steps , the value of determines the probability of to be the same as . The higher value of means the higher correlation between elements and and consequently the higher probability that will be set equal to .

Input: ,
Output: for
Main procedure
Set .
For to
Set a vector used to contain elements of a generated binary string.
Randomly select a position (th row) of .
Set . % Set the th element of a as the th element of .
For
If
% and values are equal, which are either “0” or “1”.
Else
% If , or vice versa.
End
End
Set .
End

The CBE matrix is a square symmetric matrix with equal size to the length of a binary solution whose all diagonal elements are equal to one. For an iteration, the matrix will be updated according to the so far best solution (). The learning rate () with be used to control the changes in updating as with PBIL. Once is updated, the value of is set to be which means the process requires updates since is always set to be 1. The updated denoted by can be calculated fromwhere is the learning rate randomly generated in the interval . and are the th and th elements of , respectively. From the updating equation, if the th and th elements are similar, it means they are correlated; consequently, the value of (and ) is increased. If they are dissimilar or uncorrelated, is then decreased. Nevertheless, the value of must be limited to the predefined interval where and are the predefined lower and upper limits of . Equation (3) is used to maintain diversity in optimisation search. In the original PBIL, a mutation operator is used with the same purpose. Therefore, the procedure of EDACE starts with an initial matrix for correlation between binary elements where and . This implies that when generating a binary solution, its elements have equal probability to be 1 or 0 where its th element can be 1 or 0, created at random. The procedure for general purpose of EDACE is given in Algorithm 2. The decision on selecting for generating a binary solution and for updating the CBE matrix is dependent on a preference of a user. This means other versions of EDACE can be developed in the future.

Input: number of generation (), population size (), binary length ()
Output: ,
Initialisation:
(0.1) Assign and , sized .
(0.2) Randomly generate binary solutions and decode them to be .
(0.3) Calculate objective function values where fun is an objective function evaluation.
(0.4) Find , ,
Main iterations
For to
Update using Equation (2)
Generate from using Algorithm 1, and decode them to be .
For to
Calculate objective function values .
If
, ,
End
End
Update , ,
End

An initial binary population is randomly created. The binary solutions are then decoded to be real design variables where function evaluations are performed and and are found. Then, new binary solutions are generated using Algorithm 1 while the greedy selection (steps ) is activated with and being determined. The CBE matrix is updated by using as detailed in (2)-(3). The search process is repeated until termination criterion is reached. The generation of a binary design solution of EDACE is, to some extent, similar to those used in binary PSO [49] and binary quantum-inspired gravitational search algorithm (BQIGSA) [52] in the sense that the binary solution is controlled by the probability of being “1” or “0”. However, in EDACE, a generated solution relies not only on such probability but also on the reference binary solution . Apart from that, the update of CBE tends to be similar to the concept employed in PBIL with a learning rate and this is totally different from binary PSO and BQIGSA.

In selecting and , if both solutions are the same which is , it could lead to a premature convergence. If both are set to be a solution randomly selected solution from the current binary population, the diversification increases but the convergence rate will be slower. Therefore, the balance between intensification and diversification must be made. In this work, the so far best binary solution is set to be to maintain intensification. For updating the CBE matrix, we use the new updating scheme asThe solutions and are two types of best solutions. Firstly, best solutions are selected from (see Algorithm 2 for both solution sets), sorted according to their functions, and then saved to a set Best_sol. Four vectors are created as the so far best solution, a solution whose elements are averaged from the elements of the first (default = 10) best solutions found so far, a solution whose elements are averaged from the elements of the members of Best_sol, and a solution whose elements are averaged from the elements of the current binary population. is randomly chosen from the aforementioned solutions (, , , and ) with equal probability while is randomly chosen from the members of Best_sol. With this idea, the balance between exploration and exploitation is maintained throughout the search process. Algorithm 3 shows the new CBE updating strategy.

Input: , , , , , Best_sol,
Output:
Main procedure
Create , , ,
For to
Assign .
If , set
If , set
If , set
Otherwise, set
Random selected a vector from Best_sol.
For to
Generate .
Update using Equation (4).
Limit to the interval .
End
End

3. Experimental Set-Up

To investigate the search performance of the proposed algorithm, fifteen learning-based test problems from CEC2015 and one flight dynamic control optimisation problem are used. The former are used for testing the performance of EDACE for general types of box-constrained optimisation while the latter is the real-world application.

3.1. CEC2015 Learning-Based Test Problems

The CEC2015 learning-based test problems are box-constrained single objective benchmark functions proposed in [56]. The problems consist of 2 Unimodal Functions, 3 Simple Multimodal Functions, 3 Hybrid Functions, and 7 Composition Functions. The summary of CEC2015 learning-based test problems is shown in Table 1. It should be noted that the details and the codes for the test problems can be downloaded from the website of CEC2015 competition.


NumberFunctions

Unimodal functions1Rotated high conditioned elliptic function100
2Rotated Cigar function200

Simple multimodal functions3Shifted and rotated Ackley’s function300
4Shifted and rotated Rastrigin’s function400
5Shifted and rotated Schwefel’s function500

Hybrid functions6Hybrid function 1 ()600
7Hybrid function 2 ()700
8Hybrid function 3 ()800

Composition functions9Composition function 1 ()900
10Composition function 2 ()1000
11Composition function 3 ()1100
12Composition function 4 ()1200
13Composition function 5 ()1300
14Composition function 6 ()1400
15Composition function 7 ()1500

3.2. Flight Dynamic Control Optimisation Problem

Flight dynamic control system design is a classical important application for real engineering problems. The motion of an aircraft can be described using the body axes which is herein the stability axes consisting of roll axis (), pitch axis (), and yaw axis () as shown in Figure 1. The motion of the aircraft is described by Newton’s 2nd law or equations of motion for both translational and rotational motions. The dynamical model is nonlinear but can be linearised by applying aerodynamic derivatives. Due to aircraft symmetry with respect to the plane, the linearised dynamical model can be decoupled into two groups as longitudinal motion and the lateral/directional motion. For more details of deriving the equations of motion, see [57]. In this work, only the lateral/directional motion control is considered. A state equation representing the dynamic motion of an aircraft is expressed as follows [5760]:where , is the sideslip, a velocity in direction, is the yaw rate, rate of change of rotation about the -axis, is the roll rate, rate of change of rotation about the -axis, is the bank angle, rotation about the -axis, is the kinetic energy matrix, is Coriolis matrix, is the control vector, is the aileron deflection, and is the rudder deflection.

The control vector can be expressed aswhere is a pilot’s control input vector while and are the gain matrices expressed as follows [59]:where parameters are control gain coefficients which need to be found.

From (5)-(6), the state equation for lateral/directional motion of an aircraft can be expressed as

Design optimisation of the control system of an aircraft is found to have many objectives as there are several criteria that need to be satisfied such as control stability, accuracy, sensitivity, and control effort, while the control gains coefficients are set to be design variables for an optimisation problem. In this work, the optimal flight control of an aircraft focuses on only the stability aspect. The objective function is posed to minimise spiral root subjected to stability performance constraints. The optimisation problem can then be written aswhere , , , and are spiral root, roll damping, damping ratio of Dutch-roll complex pair, and Dutch-roll frequency, respectively. These parameters can be calculated based on the eigenvalues associated with the matrix . The design variables are control gain coefficients in the matrix . The kinetic energy matrix () and the Coriolis matrix () are defined as

More details about this aircraft dynamic model can be found in [5860]. To handle the constraints, the penalty function which was presented in [61] is used.

The proposed EDACE and several well-established binary-code metaheuristics are used to solve the fifteen CEC2015 learning-based test problems and the flight dynamic control test problem. The metaheuristic optimisers are as follows:Genetic algorithm (GA) [21] used binary codes with crossover and mutation rates are 1 and 0.1, respectively.Binary simulated annealing (BSA) [50] used binary codes with exponentially decreasing temperature. The starting and ending temperature are set to be 10 and 0.001, respectively. The cooling step is set as 10.Population-based incremental learning (PBIL) [24] used binary codes with the learning rate, mutation shift, and mutation rate as 0.5, 0.7, and 0.2, respectively.Binary particle swarm optimisation (BPSO) [49] used binary codes with V-shaped transfer function while the transfer function used is the V-shaped version 4 (V4) as reported in [49]. It is noted that this version is said to be the most efficient version based on the results obtained in [49].Univariate marginal distribution algorithm (UMDA) [48] used binary codes. The first 20 best binary solutions are used to update the probability matrix.Estimation of distribution algorithm with correlation of binary elements (EDACE) (Algorithm 2) used binary codes with , , , , and .

Each algorithm is used to solve the problems for 30 optimisation runs. The population sizes are set to be 100 and 20 while number of generation is set to be 100 and 500 for the CEC2015 learning-based test problems and the flight dynamic control test problem, respectively. For an algorithm using different population size and number of generations such as BSA, it will be terminated at the same number function evaluations, which is 10,000 for all test problems. The binary length is set to be 5 for each design variable for all optimisers.

4. Optimum Results

4.1. CEC2015

After applying the proposed EDACE and several well-established binary MHs for solving the CEC2015 learning-based benchmark functions, the results are shown in Tables 24. Note that, apart from the algorithms used in this study, the results of solving CEC2015 test suit obtained from efficient binary artificial bee colony algorithm based on genetic operator (GBABC), binary quantum-inspired gravitational search algorithm (BQIGSA), and self-adaptive binary variant of a differential evolution algorithm (SabDE) as reported in [53] are also included in the comparison. From Table 2, the mean (Mean) and standard deviation (STD) values of the objective functions are used to measure the search convergence and consistency of the algorithms. The lower Mean is the better convergence while the lower STD is the better consistency. The value of Mean is more important; thus, for method A with lower Mean but higher STD than method B, method A is considered to be superior.


CEC2015MHsUMDABPSOGAPBILBSAEDACE

Unimodal functionsMean
STD
Min.
Mean
STD
Min

Simple multimodal functionsMean
STD
Min
Mean
STD
Min
Mean
STD
Min

Hybrid functionsMean
STD
Min
Mean
STD
Min